Reese picked twe numbers, \( x \) and \( y \). She told her friend that the sum of the two numbers is 9 and the difference of the two numbers is 17 . a. Write two different linear equations that model what Reese told her friend. \[ \begin{array}{l}x+y=9 \\ x-y=17\end{array} \] b. Solve the system of linear equations using the substitution method. What two numbers did Reese pick?

To solve the problem, we will follow the steps outlined: ### Part a: Write the Linear Equations Reese provided two pieces of information about the numbers \( x \) and \( y \): 1. The sum of the two numbers is 9. 2. The difference of the two numbers is 17. From this information, we can write the following linear equations: \[ \begin{array}{l} x + y = 9 \quad \text{(1)} \\ x - y = 17 \quad \text{(2)} \end{array} \] ### Part b: Solve the System of Linear Equations Using the Substitution Method We will solve the system of equations using the substitution method. 1. **From equation (1)**, we can express \( y \) in terms of \( x \): \[ y = 9 - x \quad \text{(3)} \] 2. **Substitute equation (3)** into equation (2): \[ x - (9 - x) = 17 \] 3. **Simplify the equation**: \[ x - 9 + x = 17 \] \[ 2x - 9 = 17 \] 4. **Add 9 to both sides**: \[ 2x = 26 \] 5. **Divide by 2**: \[ x = 13 \] 6. **Now substitute \( x = 13 \)** back into equation (3) to find \( y \): \[ y = 9 - 13 \] \[ y = -4 \] ### Conclusion The two numbers that Reese picked are: \[ x = 13 \quad \text{and} \quad y = -4 \]